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Let A n = ∫ tan n ⁡ x d x ,   ∀ n ∈ N . If A 10 + A 12 = tan m ⁡ x m + λ (where λ is an arbitrary constant), then the value of m is equal to

Options

  1. A10
  2. B11
  3. C12
  4. D13

Correct answer

B. 11

Step-by-step solution

A 10 + A 12 = ∫ tan 10 ⁡ x d x + ∫ tan 12 ⁡ x d x = ∫ tan 10 ⁡ x + tan 12 ⁡ x d x = ∫ tan 10 ⁡ x 1 + tan 2 ⁡ x d x = ∫ tan 10 ⁡ x ⋅ sec 2 ⁡ x d x Let, tan ⁡ x = t ⇒ sec 2 ⁡ x d x = d t ∴ A 10 + A 12 = ∫ t 10 d t = t 11 11 + λ = tan 11 ⁡ x + λ 11

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